Indeed, it is not hard to see that it is equivalent to the Boolean prime ideal theorem ( BPIT ), a well-known intermediate point between the axioms of Zermelo-Fraenkel set theory ( ZF ) and the ZF theory augmented by the axiom of choice ( ZFC ).

Indeed, a Boolean variable may be regarded ( and be implemented ) as a numerical variable with a single binary digit ( bit ), which can store only two values.

Indeed, the classical Boolean conjunction is both commutative and associative.

Indeed, they are formally defined as the algebra of bounded operators leaving invariant each subspace contained in a subspace nest, that is, a set of subspaces which is totally ordered by inclusion and is also a complete lattice.

Indeed, the compact operators on a Hilbert space form a maximal ideal, so the quotient algebra, known as the Calkin algebra, is simple.

Indeed, if a simple module is projective, then it is the only simple module in its block, which is then isomorphic to the endomorphism algebra of the underlying vector space, a full matrix algebra.

Indeed television sets sold throughout the 1970s and early 1980s had a spare channel called ' ITV / IBA 2 '.

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

Indeed, in the U. S. estimates of the environmental regulation's total costs reach 2 % of GDP, and any new regulation will arguably contribute in some way to that burden.

Indeed, Emperor Louis Napoleon III surrendered himself and 100, 000 French troops to the German troops at Sedan on 1 – 2 September 1870.

Indeed, if x and y are integers satisfying this equation, then x / y is an approximation of √ 2.

Indeed, 2, 4-D is not a poison, but classified as " harmful " ( EU ).

Indeed 78. 2 ° C is the minimum temperature at which any ethanol / water solution can boil at atmospheric pressure.

Indeed, phosphorylation replaces neutral hydroxyl groups on serines, threonines, or tyrosines with negatively-charged phosphates with pKs near 1. 2 and 6. 5.

Indeed, it was under his orders that Lt. Francisco de Castañeda went to Gonzales, Texas, to secure a cannon from the hands of the Texans, leading to the battle on October 2 that initiated hostilities in the Texas Revolution.

Indeed, nearly 67 % of the haplogroups resulting from are Euro – Eurasian ( R1a ( 2. 76 %), R1b ( 50. 62 %), I ( 9. 66 %) and G ( 3. 99 %)).

Indeed, the center consists solely of the identity element and x < sup > 2 </ sup >.

Indeed, in 1969 a " Birding Glossary " appeared in Birding Volume 1, No. 2, which stated the following:

Indeed no communication was ever established and Beagle 2 was declared lost on 6 February 2004.

Indeed, on the evidence, I find that the public want to be able to buy tea and coffee served hot, that is to say at a temperature of at least 65 ° C, even though they know ( as I think they must be taken to do for the purposes of answering issues ( 1 ) and ( 2 )) that there is a risk of a scalding injury if the drink is spilled.

Indeed, the inland and coastal populations of A. californica seem to constitute 2 distinct species too, as might different populations of the Mexican Jay.

Indeed, the full Bible has been translated into over 450 languages, and some sections of the Bible have been translated into over 2, 000 languages.

Indeed, our whole knowledge of God is limited to two certainties, ( 1 ) that He exists and ( 2 ) that His essence is incomprehensible ( ib.

Indeed, God " desires all men to be saved and to come to the knowledge of the truth " ( 1 Timothy 2: 4 ); that is, God wills the salvation of everyone through the knowledge of the truth.

Indeed, C has one eigenvalue ( namely zero ) and this eigenvalue has algebraic multiplicity 2 and geometric multiplicity 1.

Indeed, the UN declaration of partition, though it was generally met with by peaceful protests in Arab countries, provided the pretext for a pogrom in Aden soon after, on 2 December 1947, in which 82 Jews were killed.

Indeed each incidence structure gives a spherical building of rank 2 ( see ); and Ballmann and Brin proved that every 2-dimensional simplicial complex in which the links of vertices are isomorphic to the flag complex of a finite projective plane has the structure of a building, not necessarily classical.

Indeed, he obtained 49. 2 % of votes in the second round.

Indeed, in the case of Kirmani v Captain Cook Cruises Pty Ltd ( No 2 ) ( 1985 ), the court said that it would never again grant a certificate of appeal.

Indeed, the area of a disk of radius R is πR < sup > 2 </ sup > and the circumference of the circle is 2πR, so both sides of the inequality are equal to 4π < sup > 2 </ sup > R < sup > 2 </ sup > in this case.

Indeed, a lighter weight works much better because a greater, more extensive split can be performed.

( Indeed, the set of endings can be replaced by the name of a set of endings.

Indeed, one can assume that exactly one parallel through a point outside a line exists, or that infinitely many exist.

Indeed, the International Committee for Weights and Measures ( CIPM ) notes that " its definition applies only within a spatial extent sufficiently small that the effects of the non-uniformity of the gravitational field can be ignored.

Indeed, Disraeli had objected to Murray about Croker inserting " high Tory " sentiment, writing that " it is quite impossible that anything adverse to the general measure of Reform can issue from my pen.

Indeed, one such clause can be rewritten as, that is, if are all true, then y needs to be true as well.

Indeed it can be argued the country has never been independent since there is an arguable legitimate succession of states, systems and entities from the Norman Conquest, 1066.

Indeed, it turns out that numerous important constructions can be described in a purely categorical way.

( Indeed, every production of a context free grammar is of the form V → w where V is a single nonterminal symbol, and w is a string of terminals and / or nonterminals ( w can be empty )).

Indeed, the symbiosis with root fungi is quite common among the order representatives, and there are even three kinds of it which can be found exclusively among Ericales ( namely, ericoid, arbutoid and monotropoid mycorrhiza ).

Indeed, Winograd showed that the DFT can be computed with only irrational multiplications, leading to a proven achievable lower bound on the number of multiplications for power-of-two sizes ; unfortunately, this comes at the cost of many more additions, a tradeoff no longer favorable on modern processors with hardware multipliers.

In a ring all of whose ideals are principal ( a principal ideal domain or PID ), this ideal will be identical with the set of multiples of some ring element d ; then this d is a greatest common divisor of a and b. But the ideal ( a, b ) can be useful even when there is no greatest common divisor of a and b. ( Indeed, Ernst Kummer used this ideal as a replacement for a gcd in his treatment of Fermat's Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or ideal, ring element d, whence the ring-theoretic term.

Indeed it can be shown that any Q-automorphism must preserve the ordering of the real numbers and hence must be the identity.

Indeed, the importance of the concepts of good and evil can be regarded as distinguishing mark between high fantasy and sword and sorcery.

Indeed, an intuitive summary of inertial frames can be given as: In an inertial reference frame, the laws of mechanics take their simplest form.

Indeed, the entrenched nature of command-and-control can be seen in the way that recent layoffs have been conducted with management ranks affected far less than employees at the lower levels.

Indeed, this can be extended to the observation that introducing a DNA-binding sequence within the nucleosome increases the accessibility of adjacent regions of DNA when bound.

Indeed, he argues that any course of action can be made out to accord with some particular rule, and that therefore a rule cannot be used to explain an action.

Indeed, the renowned physicist Richard Feynman once said, " I think I can safely say that nobody understands quantum mechanics.

Indeed Ingram's influence on Powell's later work can be detected, especially in its themes in illusion, dreaming, magic and the surreal.

Indeed, the scientific method, and the scientists themselves, can be prepared in many other ways to harness luck and make discoveries.

* " Indeed men too often take upon themselves in the prosecution of their revenge to set the example of doing away with those general laws to which all can look for salvation in adversity, instead of allowing them to subsist against the day of danger when their aid may be required " ( Strassler 201 / 3. 84. 3 ).

Indeed Fermat's principle does not hold standing alone, we now know it can be derived from earlier principles such as Huygens ' principle.

i. e., z is a vector orthogonal to the vector v ( Indeed, z is the projection of u onto the plane orthogonal to v .) We can thus apply the Pythagorean theorem to